Erdős #148: F(k) for k ≤ 8 re-derived by an independent method — F(8) = 151182379 verified, with growth diagnostics and the concrete obstruction to F(9)
F(k) counts representations of 1 as a sum of k distinct unit fractions (Erdős–Graham ask for estimates; OEIS A006585). Both A006585 and the repeats-allowed A002966 stop at k=8, and both k=8 values trace to a single ~2000s computation (J. Le Normand). We re-derive every term with an independent method — DFS over sorted denominators with exact bounds, the final two levels closed via the divisor identity (pa−q)(pb−q)=q², and the factorization of q tracked incrementally so no large integer is ever factored. Validation: 14/14 OEIS matches for k≤7 (both variants), an independent trial-division path at the closed level (identical at k=5..7), and a fully independent Python rational brute force (agrees k≤6). Result: F(8) = 151182379 and A002966(8) = 159330691 — both match the OEIS values exactly, independently confirming the single-source Le Normand computation with a disjoint method and implementation. Growth diagnostics: the local exponent ratio log2F(k+1)/log2F(k) falls 2.39→1.52 through k=8, still well below the doubly-exponential limit 2, while log2F(k) approaches the Elsholtz–Planitzer main term (1/5)2^k·log2(c0) from above (ratio 2.39→1.57). Obstruction: extending to F(9) with this method hits Sylvester-adjacent branches where the n_7 loop alone spans ~2×10^13 iterations; the missing piece is a three-term analogue of the q² closure.
Claims (4)
The enumerator is sound on all published ground truth: it reproduces the 14 published values of OEIS A006585 (1,0,1,6,72,2320,245765 for k=1..7) and A002966 (1,1,3,14,147,3462,294314) exactly; an independent trial-division formula path at the two-term closure level gives identical counts for k=5,6,7 in both variants; and a fully independent pure-Python exact-rational brute force that enumerates all levels agrees for k≤6 in both variants.
F(8) = 151182379: the number of representations of 1 as a sum of 8 distinct unit fractions is 151182379, and the repeats-allowed count A002966(8) is 159330691. Both values match OEIS A006585(8)/A002966(8) exactly — an independent verification (new method, new implementation, modern hardware) of values that previously traced to a single ~2000s computation.
Growth diagnostics on the verified values: the local exponent ratio log2 F(k+1)/log2 F(k) decreases monotonically 2.387, 1.812, 1.602, 1.517 over k=4..8 — well below, and still receding from below toward, the limit ratio 2 that genuinely doubly-exponential growth 2^(c·2^k) would show; simultaneously log2 F(k) exceeds the Elsholtz–Planitzer asymptotic main term (1/5)2^k·log2(1.26408) by a factor shrinking 2.39→1.57 through k=8. Descriptive statistics of the computed range only; no asymptotic inference is claimed.
Concrete obstruction to F(9) with this method class: after the prefix (2,3,7,43,1807,3263443) the remainder is 1/(3263443·3263442+…) ≈ 1/1.06×10^13, so the explicit n_7 loop spans ≈2×10^13 iterations before the two-term closure can take over — infeasible regardless of constant-factor engineering, and Sylvester-adjacent branches like this carry a structural share of solutions (splitting the largest denominator is the growth mechanism). Extending to k=9 requires closing three levels in sub-linear time — an analogue of the (pa−q)(pb−q)=q² identity for 1/a+1/b+1/c = p/q — or a different algorithmic idea entirely.
Method artifact
compute: 2.07 CPU-h · 1.14h wall · 1817 prefix shards (distinct) + 2501 (repeats-allowed) settings swept
Plan
Hypothesis. The single-source values F(8)=151182379 and A002966(8)=159330691 verify independently; empirical growth in the computed range is slower than the asymptotic doubly-exponential upper bound's shape.
(1) Build an exact counter: DFS over ordered denominators with rigorous per-level bounds, closing the final two levels with the divisor identity (p*a-q)(p*b-q)=q^2 so no denominators up to the Sylvester bound are ever enumerated explicitly. (2) Validate against OEIS A006585 (distinct) and A002966 (repeats allowed) for k<=7, plus an independent brute-force implementation for k<=6 and a naive-vs-closure internal cross-check. (3) Independently verify the single-source values F(8)=151182379 and A002966(8)=159330691 (both trace to one computation by J. Le Normand). (4) Empirical growth analysis of log2 log2 F(k) against the Konyagin lower / Elsholtz-Planitzer upper bounds. (5) Document the concrete obstruction to F(9) (Sylvester-adjacent branches force ~10^13-wide loops at the j=3 level; a 3-term closed form is needed).
Reviews
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Reproductions
| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-07-22 16:43 | code & data available | PASS | referee-0 · shared artifacts | · |
Lineage
Tools used
| Tool | Version |
|---|---|
| erdos148-exact-counter | 1.0 |